Proc. Natl. Acad. Sci. USA
Vol. 95, pp. 201–206, January 1998
Cell Biology
Control of direction of flagellar rotation in bacterial chemotaxis
(Escherichia coliysignal transductionyCheYymolecular motors)
BIRGIT E. SCHARF*, KAREN A. FAHRNER*, LINDA TURNER*†,
AND
HOWARD C. BERG*†‡
*Department of Molecular and Cellular Biology, Harvard University, Cambridge, MA 02138; and †The Rowland Institute for Science, Cambridge, MA 02142
Contributed by Howard C. Berg, November 13, 1997
ABSTRACT
The motile behavior of the bacterium Escherichia coli depends on the direction of rotation of its flagellar
motors. Binding of the phosphorylated signaling molecule CheY
to a motor component FliM is known to enhance clockwise
rotation. It is difficult to study this interaction in vivo, because the
dynamics of phosphorylation of CheY by its kinase CheA and the
hydrolysis of CheY (accelerated by CheZ) are not under direct
experimental control. Here, we examine instead the interaction
with the flagellar motor of a double mutant CheY13DK106YW that
is active without phosphorylation. The behavioral assays were
carried out on tethered cells lacking CheA and CheZ. The effects
of variation in intracellular concentration of the mutant protein
were highly nonlinear. However, they can be explained by a
thermal isomerization model in which the free energies of
clockwise and counterclockwise states depend linearly on the
amount of CheY bound.
MATERIALS AND METHODS
Bacteria, Phage, and Plasmids. (See Table 1.) The following
were gifts: pXYZ202 from Xiangyang Zhu and Phil Matsumura,
EC0 and phage f1R408 from Jon Beckwith, pBIP from Steven
Slater, MM5008 from Mike Manson, pACYC184-Iq from Karen
McGovern, pBR322yhag93 and pFD313 from Goro Kuwajima,
and RP9535 and RP4979 from Sandy Parkinson.
pSE420 contains an initiating ATG at its unique NcoI site
(base pair 412) at the start of the 353-bp SuperLinker (SL2).
This translational start site was eliminated by restriction with
StyI, which recognizes both the NcoI site and a downstream
StyI site at base pair 945, and religation of the larger 4380-bp
fragment. The resulting plasmid, pSE420DStyI, was checked
for the loss of the NcoI site.
A genomic region containing part of cheR, all of cheB and cheY,
and part of cheZ was obtained from a pool of 1.6- to 2.0-kb
MluI-restricted DNA from HCB758. These fragments were
cloned into the MluI site of pSE280, and the resulting plasmids
were transformed into the cheY deletion strain RP4979. Transformants were tested for restoration of swarming on soft agar
plates (0.3% agary1% tryptoney0.5% NaCl). One complementing plasmid, pKAF118, was obtained. The MluI genomic insert
was recloned in the reverse orientation, giving pKAF119.
Cross-In Construction. A construct was assembled in
pBluescript IISK(1) (pBES32) that contains the following:
cheY13DK106YW under control of the isopropyl b-D-thiogalactoside (IPTG)-inducible promoter, Ptrc; a deletion in the adjacent gene, cheZ; the cat (chloramphenicol transacetylase)
gene; and DNA flanking the cheY locus to provide homology
for crossing in. This construct, pBES32, is shown in Fig. 1. Its
constituent parts, sources, and the restriction sites used in
assembly are listed in Table 2. When necessary for ligation,
restricted DNA fragments with overhangs were blunt-ended by
treatment with T4 DNA polymerase andyor E. coli DNA
Polymerase I Large Fragment (New England Biolabs). In
addition to Ptrc, the construct contains the following elements
derived from pSE420: the lac operator (lacO), a sequence from
rrnB that inhibits premature termination, and the bacteriophage T7 gene 10 mini-cistron with an internal ribosomebinding site and translation termination codon. The ability of
pBES32 to induce CW rotation with added IPTG was tested
in RBB1041, a strain deleted for chemotaxis genes cheA
through cheZ. The insert of pBES32 contains flanking NotI
sites and was inserted into the NotI site of the bacterial
integrating plasmid, pBIP, to give pBES36.
Strain Construction. The cheY region of RP9535, a strain with
a preexisting cheA deletion, was replaced with that constructed in
pBES36 by recombination by using the phagemid-based scheme
of Slater and Maurer (16). In this method, transfer is accomplished by infection with recombinant filamentous phage; there-
Motile bacteria such as Escherichia coli actively respond to a
variety of stimuli by modulating the direction of rotation of
their flagella. Addition of a chemical attractant (e.g., aspartate) or removal of a chemical repellent (e.g., leucine) enhances counterclockwise (CCW) rotation, causing cells to
extend runs that carry them in a favorable direction. This
occurs through regulation of a kinase, CheA, which phosphorylates an effector molecule, CheY. Phospho CheY (CheY-P)
when bound to a component at the base of the flagellar motor,
FliM, promotes clockwise (CW) rotation (for reviews, see refs.
1–4). This interaction has been studied in vivo (5), but interpretation of the results is complicated by the dynamics of
phosphorylation and hydrolysis: the intracellular concentration of CheY-P was not measured.
One solution to this problem is to use CheY mutants that are
active without phosphorylation. Wild-type CheY (CheYwt) is
activated by phosphorylation at Asp-57 (6). Replacement of
Asp-13 by Lys (or Arg) results in a CW phenotype (7).
CheY13DK can be phosphorylated to some extent, but phosphorylation is not required for activity. Replacement of Tyr106 by Trp results in an even stronger CW phenotype (8), but
only when CheY106YW is phosphorylated. The double mutant
CheY13DK106YW—we will call this protein CheY**—is active
without phosphorylation (X. Zhu and P. Matsumura, private
communication). We chose this mutant to study the dynamics
of the interaction of CheY with the flagellar motor.
There are two kinds of mechanisms that might explain how
CheY controls the direction of the flagellar motor. In one, the
switch is thrown when a certain number of CheY molecules are
bound. In the other, the number bound only determines the
probability of CW or CCW rotation, and the switch is thrown
by thermal fluctuations. Our results argue for the latter,
stochastic mechanism.
Abbreviations: CheY**, CheY13DK106YW; CheYwt, wild-type CheY;
CheY-P, phospho CheYwt; CW, clockwise; CCW, counterclockwise;
ECL, enhanced chemiluminescence; FliCst, flagellin yielding sticky
filaments; IPTG, isopropyl b-D-thiogalactoside.
‡To whom reprint requests should be addressed at: Bio Labs, Harvard
University, 16 Divinity Ave., Cambridge, MA 02138. e-mail:
[email protected].
The publication costs of this article were defrayed in part by page charge
payment. This article must therefore be hereby marked ‘‘advertisement’’ in
accordance with 18 U.S.C. §1734 solely to indicate this fact.
© 1998 by The National Academy of Sciences 0027-8424y98y95201-6$2.00y0
PNAS is available online at http:yywww.pnas.org.
201
202
Cell Biology: Scharf et al.
Table 1.
Bacterial strains and plasmids used in this study
Strain or plasmid
Strains
DH5a
EC0
HCB5*
HCB758*
HCB899†
HCB900
HCB901
HCB902
HCB1264*
JM109
MM5008†
RBB1041†
RP3098
RP4979†
RP9535†
Plasmids
pACYC184-Iq
pBES32
pBES36
pBES38
pBIP
pBSIISK(1)‡
pBR322yhag93
pFD313
pKAF118
pKAF119
pMAK705
pRL22
pSE280
pSE420
pSE420DStyI
pXYZ202
Proc. Natl. Acad. Sci. USA 95 (1998)
Relevant genes
Reference or source
Remarks
recA1 endA1 gyrA96
F9ts 114 lac1yD(lac-proB)XIII
fliC726
wild type for chemotaxis
F9ts 114 lac1yD(cheA)1643
F2 D(cheA)1643 cat Ptrc420 cheY13DK106YW DcheZ
GIBCOyBRL
9
P. Conley
10
This work
This work
F2 D(cheA)1643 cat Ptrc420 cheY13DK106YW DcheZ
fliC726 uvrC-279::Tn10
F2 D(cheA)1643 cat Ptrc420 cheY13DK106YW DcheZ
fliC726 uvrC-279::Tn10ypBES38
fliC726 uvrC-279::Tn10
F9 traD36yrecA1
uvrC-279::Tn10
D(cheA-cheZ)::ZeoR
This work
Host for plasmid propagation
Source of F9ts
Source of fliC null mutation
Source of MluI genomic fragment
F9ts derivative of RP9535
DcheZ derivative of HCB899 with
IPTG-inducible cheY13DK106YW
Nonmotile derivative of HCB900
This work
lacIq, fliCst derivative of HCB901
This work
11
12
W. N. Abouhamad
and R. B. Bourret
13
J. S. Parkinson
14
Tn10-linked fliC726 derivative of HCB5
Host for f1R408
Source of Tn10 marker near fliC
Host for testing cross-in constructs
D( flhA-flhD)
D(cheY)m43-10
D(cheA)1643 (eda1) lacY1
tet lacIq
bla 9cheR cheB cat cheY13DK106YW 9cheZ
npt sacB 9cheR cat cheY13DK106YW 9cheZ
bla fliCst lacIq
npt sacB
bla
bla fliCwt
bla fliCst
bla 9cheR cheB cheY cheZ9
bla 9cheR cheB cheY cheZ9
cat
bla 9cheB cheY cheZ flhB9
bla
bla
bla
bla cheY13DK106YW
15
This work
This work
This work
16
Stratagene
17
18
This work
This work
19
20
Invitrogen
Invitrogen
This work
X. Zhu and
P. Matsumura
cheY, cheZ deleted strain
Host for selection of pKAF118
Background DcheA strain
Source of lacIq (1.1 kb EcoRI insert)
Cross-in construct in pBSIISK(1)
Cross-in construct in pBIP
Provides LacI and FliCst in HCB902
Phagemid-based integrating plasmid
Cloning vector
Source of FliCwt in pBR322
Source of FliCst in pBR322
Genomic MluI insert in pSE280
Reversed MluI insert of pKAF118
Source of cat gene
Source for construction of DcheZ
Cloningyexpression vector
Cloningyexpression vector
pSE420 without transcriptional start site
Source of cheY13DK106YW
*Derivative of AW405 (21).
†Derivative of RP437 (22).
‡pBluescript IISK(1).
fore, an F factor was introduced into RP9535 (cheA lacY1) by
mating with EC0, a Pro2 strain that contains a temperaturesensitive F factor that carries the lac operon, and selecting for the
ability to use lactose as a carbon source in the absence of proline,
FIG. 1. The segment of DNA (Lower) crossed into the chromosome of a cheA deletion strain (Upper) to yield strain HCB900, drawn
to the same scale. The shaded regions are regions of homology used
in the cross-in. cheY was replaced by cheY13DK106YW and most of cheZ
was deleted. Base pairs are numbered and fragments are labeled as in
Table 2. Arrows, transcription units; arrowheads, hybridization sites
for primers; open boxes and lines, genomic DNA; solid boxes, lines,
and circle, plasmid DNA.
at 30°C. The resulting strain, HCB899, was infected with the
helper phage strain, f1R408, which had been grown on JM109
transformed with pBES36. f1R408 preferentially packages ss
plasmid DNA (23). Recombinants were selected for resistance to
kanamycin (50 mgyml), followed by a second selection for resolved products by resistance to both chloramphenicol (10 mgy
ml) and sucrose (5%), as described (16). The resulting strain, now
kanamycin-sensitive, was passaged at 42°C to promote loss of the
episome. This strain, HCB900, proved negative for infection by
M13 (no plaque formation).
To confirm the deletion of the cheZ gene, PCR amplification of HCB900 DNA by using flanking primers 64R (59ACACCGGCTTTGCTGGTATC-39) and 68F (59-GTTATGGATTTGTTATCTCCGAC-39) (Fig. 1) generated a fragment
of the expected 315-bp size, compared with a fragment of 844
bp generated by parental strain DNA. Furthermore, no CheZ
could be detected in immunoblots by using anti-CheZ mAb
(see below). To confirm the transfer of cheY13DK106YW, PCR
amplification by using the primer pair 69R (59-GTGGTATTCACTCCAGAGCG-39) and 64R generated a single fragment of the expected '1.3 kb, compared with the absence of
any fragment generated by the parental strain DNA.
To facilitate analysis of the switching behavior of tethered
cells, we utilized the self-tethering property of flagellar filaments composed of an internally truncated flagellin (24). First,
Cell Biology: Scharf et al.
Table 2.
Proc. Natl. Acad. Sci. USA 95 (1998)
203
Segments comprising NotI insert of pBES32 (and pBES36)
Segment*
Nucleotide
number†
Source‡
Restriction site§
A
B
C
D
E
F
G
H
I
J
K
1–95
96–1414
1415–1431
1432–2582
2583–2592
2593–3368
3369–3817
3818–3882
3883–3905
3906–4916
4917–4941
pSE280 (456–550)
pKAF119
pBSIISK(1)
pMAK705
pBSIISK(1)
pSE420DStyI (4536–699)
pXYZ202
pSE420DStyI (702–759)
pSE280 (331–352)
pRL22
pBSIISK(1)
KpnI-MluI
MluI-(DraI)
(HincII)-(HindIII)
(Bsu36I)-(AatII)
(HindIII)-(EcoRV)
(PvuII)-(NruI)
(DraI)-(BsmI)
(NruI)-(HindIII)
(SalI)-(SmaI)
(AvaII)-BamHI
BamHI-NotI
*Segments, as labeled in Fig. 1. A, C, E, H, I, and K are fragments of polylinkers. B, G, and J are che
coding regions containing 39(cheR)-cheB, cheY13DK106YW, and 39(cheZ)-59( flhB), respectively. D is a
fragment containing cat, and F comprises regulatory elements derived from pSE420 (see Section titled
Cross-In Construction).
†Deduced from published sequences and expected results of ligations.
‡Numbers in parentheses are nucleotide number of source.
§Restriction sites used to generate fragment from source. Parentheses indicate that the original site was
not regenerated after ligation.
a nonreverting null mutation, fliC726 (25), was introduced into
HCB900 by P1 transduction, giving HCB901. Then, pFD313,
which encodes a mutant flagellin in which 57 centrally located
residues are replaced by 6 other residues (18), referred to as
FliCst, was transformed into HCB901.
Preliminary swimming assays suggested that expression of
CheY** in uninduced cells of HCB901 transformed with
pBR322yhag93 was high enough to cause significant switching.
To increase the range of inducibility, an EcoRI fragment encoding LacI from pACYC184-Iq was inserted into the unique EcoRI
site of pFD313, and the resulting plasmid, pBES38, was transformed into HCB901, giving the final strain, HCB902. This was
effective in reducing the uninduced concentration of CheY**, as
confirmed by comparative immunoblots.
Purification of CheYwt, CheY**, and CheZ. Proteins were
isolated essentially as described (26) by using the host strain
RP3098 (DflhA-flhD). CheYwt and CheZ were produced from
plasmid pRL22, and CheY** was produced from plasmid
pXYZ202. Concentrations of purified proteins were determined
from the Ala and Phe contents by amino acid analysis performed
by the Microchemistry Facility, Bio Labs, Harvard University.
Purification of mAb. Clones of mouse mAb against CheYwt
and CheZ were generated at the Max-Planck-Institut für
molekulare Physiologie, Dortmund, Germany, and propagated
in DMEMy10% fetal bovine serumy10% FetalClone I (HyClone) with 50 mgyml gentamycin sulfate (GIBCOyBRL).
Antibodies were purified from tissue culture supernatants by
ammonium sulfate precipitation, followed by chromatography
on Protein G Sepharose 4 (Pharmacia) according to the
manufacturer’s instructions. Protein concentrations were determined with Bio-Rad Protein Assay.
Cell Cultures. Cells were prepared in the same way for
quantitative immunoblots and behavioral assays. Strain HCB902
was grown overnight from frozen stocks in a 125-ml culture flask
containing 10 ml tryptone broth (TB) (1% tryptoney0.5% NaCly
0.1% yeast extracty100 mg/ml ampicillin). Saturated cultures were
diluted 1:200 in a 250-ml flask containing 40 ml TB and 100
mgyml ampicillin. After 4 h, IPTG from a freshly thawed 0.01 or
0.1 M stock solution was added and incubation was continued for
2 h, to OD610 ' 0.8. All incubations were at 33°C and 200 rpm.
Strain RP3098 was grown as HCB902, omitting ampicillin and
IPTG. Actual cell densities were determined by viable cell counts
of serially diluted cultures plated on Luria–Bertani (LB) agar
plates. Because cells of HCB902 stick to glass and yield an
artificially low count, cells of HCB901 transformed with pBR322y
hag93 were used for this determination. At OD610 5 0.8 (mea-
sured with Hitachi spectrophotometer U-3000), cell density was
(6.2 6 0.7) 3 108 cells per ml.
Dry weights of cells were determined as follows. Cells were
grown as above. Four 30-ml samples were harvested by centrifugation, resuspended in 77 mM ammonium acetate, pH 7.0,
transferred to tared Eppendorf tubes, centrifuged, washed in
the same buffer, and lyophilized for 2–3 days. Medium and
buffer were prefiltered (0.2 mm). A value of 0.26 6 0.01 mgyml
of culture (mean 6 SD for five determinations) was obtained.
Quantitative Immunoblots. Aliquots of strain HCB902 (0.2–
1.0 ml) were used with aliquots of strain RP3098 to bring the final
volume to 1.0 ml. Cells were harvested by centrifugation, washed
once in 50 mM TriszHCly0.5 mM EDTAy2 mM DTTy10%
(vol/vol) glycerol, pH 7.5, resuspended in 40 ml of SDS–sample
buffer, and heated to 100°C for 6 min. Samples of this mixture
(called cell extracts) were stored at 220°C. Control samples were
prepared by adding 10–60 ng of purified CheY** to an extract
prepared from a 1-ml aliquot of RP3098. Four samples and nine
standards (at three different concentrations) were applied to each
gel. Electrophoresis was performed by the procedure of Laemmli
(27) in a 1-mm-thick gel in a linear gradient from 12.5 to 20%
acrylamide (National Diagnostics). Electrophoretic transfer of
proteins from gels to 0.45 mm nitrocellulose (Hybond ECL,
Amersham) was done in a tank blot device (Bio Labs, Harvard
University) for 1.5 h at 500 mA by using Bjerrum and Schafer–
Nielsen buffer (28). Nitrocellulose blots (4 cm 3 14 cm) were
blocked overnight at room temperature in 80 mM Na2HPO4y20
mM NaH2PO4y100 mM NaCly0.1% (volyvol) Tween 20 (BioRad), pH 7.5y5% instant nonfat dry milk (Carnation) on a
rocking platform. The remaining steps were carried out in the
same medium in a similar manner. Blots were probed with 0.01
mgyml anti-CheYwt mAb for 3 h, washed three times (10 min
each), incubated for 2 h with sheep anti-mouse horseradish
peroxidase-linked whole Ig antibody (Amersham) diluted 1:1000,
and washed four times (10 min each). The detection reaction, by
using enhanced chemiluminescence (ECL, Amersham), was performed according to the manufacturer’s protocol. Preflashed
Hyperfilm ECL (Amersham) was used for detection with a series
of exposures. Films were scanned in the presence of a calibrated
Step Tablet (Eastman Kodak) by using a Scan Jet IIcxyT (Hewlett–Packard) and DESKSCAN II 2.3 software. Analysis of scans was
performed by using NIH IMAGE 1.59 and KALEIDAGRAPH 3.8.
Tethering. A glass coverslip silanized with Rain-X (Unelco,
Scottsdale, AZ) was supported over a glass slide at its edges by
two other coverslips, which were greased together with Apiezon L (Fisher). Cells were sheared 45 times (29) and added
204
Cell Biology: Scharf et al.
to the space between the coverslip and the slide. The slide was
inverted and, after 3–5 min, inverted again and rinsed with
several volumes of 10 mM potassium phosphate (pH 7.0)y67
mM NaCly0.1 mM EDTA. Data were collected for up to 2 h.
Data Acquisition and Analysis. The angular positions of tethered cells, rotating under the coverslip, were measured individually with a linear-graded filter apparatus (30), which generates
two output voltages, x and y, proportional to the cosine and sine
of the angle of the cell image. These output voltages were
low-pass filtered (two-pole filter, time constant, 4.7 ms) and
sampled 256 times per second, in blocks of 128 s, by using
LABVIEW 3.1 (National Instruments, Austin, TX). The number of
blocks per cell varied from 2 to 10. The raw data were smoothed
by averaging successive data points, and an envelope-detection
routine was used to eliminate DC offset and correct for drift. This
routine normalized the peak values of the cosine and sine
functions to 1. The angle u was calculated as arctan(sinycos).
Rotation speeds, n 5 (1/2p)duydt, were calculated as n(i) 5 256
(1/4p) [u(i11) 2 u(i 2 1)] and then smoothed by applying a
median filter of rank 2. Speed records resembled square waves,
with positive values for CCW rotation intervals and negative
values for CW rotation intervals (Fig. 3A Inset). Speeds ranged
from 4 to 20 Hz. The CW bias (the fraction of time a cell spun
CW) was calculated as the number of data points with negative
values divided by the total number of data points, for all blocks.
The reversal frequency was calculated from the number of zero
crossings divided by the duration of the blocks. A two-state model
was assumed, and the forward (CCW-to-CW) and reverse (CWto-CCW) rate constants were calculated as k1 5 reversal frequencyy[2(1 2 bias)] and k2 5 reversal frequencyy[2(bias)],
respectively (29, 31). For cells with a sufficient number of
reversals (80 or more), these rate constants also were estimated
from exponential fits to distributions of CCW and CW interval
lengths. Intervals were not counted at the beginning or the end
of a block, where initial and final reversal times were not known.
Bias CW was computed from k1y(k1 1 k2), and reversal
frequency was computed from 2 k1 k2y(k1 1 k2).
RESULTS
Strain HCB902. This strain carries in single copy
cheY13DK106YW under control of the promoter Ptrc (inducible by
IPTG) as a replacement for wild-type cheY, on a chromosome
deleted for cheA and cheZ (Fig. 1). CheY13DK106YW (CheY**)
promotes clockwise flagellar rotation in the absence of phosphorylation (X. Zhu and P. Matsumura, private communication). Its expression is enhanced by inclusion of genetic
elements that improve the efficiency of transcription and
translation and tightened by expression, from the plasmid
pBES38, of repressor LacI. The plasmid also carries the gene
for a flagellin that forms sticky filaments, obviating the need
for antifilament antibody in tethering.
CheY** Induction. The amount of CheY** produced by
HCB902 at various levels of IPTG induction was determined by
immunoblotting by using anti-CheYwt mAb 1E7B11 and ECL
detection. The response of this system was nonlinear to both
purified CheY** and CheYwt. Consequently, all samples were
chosen to contain an optimum range of CheY** (between 10 and
60 ng), amounts that gave the steepest linear responses. The assay
also was sensitive to the total amount of protein in each sample,
so this was held constant by addition of extracts from a congenic
strain deleted for most of flagellar region II. The induction profile
is shown in Fig. 2. About 2,500 molecules per cell are expressed
in uninduced cultures, and this number is doubled at about 17 mM
IPTG.
Behavioral Response. The CW bias and reversal frequency of
tethered cells observed at different levels of CheY** induction
are shown in Fig. 3, together with the forward and reverse rate
constants k1 and k2. The scatter is a result of variations from cell
to cell and was largest for reversal frequency. The Inset in Fig. 3A
is a short segment of a speed record, n(t), of the type from which
Proc. Natl. Acad. Sci. USA 95 (1998)
FIG. 2. Induction of CheY**. At least six immunoblots (24 samples) were analyzed for each induction level. The error bars are
standard deviations of the means of each immunoblot. The line is a
polynomial fit. Ng of CheY** were determined at each IPTG level and
the molecules per cell were computed from the molecular mass (14.3
kDa) and the measured cell density (6.2 3 108 cells per ml). Corresponding concentrations were computed from the measured dry
weightyml cell culture (0.26 mg) and the cytoplasmic volumeymg dry
weight (1.4 ml, ref. 32).
directions of rotation were determined. The values of the rate
constants deduced from measurements of bias and reversal
frequency agreed closely with those made from exponential fits
to interval distributions. This agreement, together with the fact
that the distributions were exponential (data not shown), support
the view that the motor is essentially a two-state system, with
switching probabilities per unit time (k1 and k2) that depend on
the amount of CheY** bound. To a first approximation, the
impact of larger levels of CheY** is symmetric: k1 rises and k2
falls; their values are the same at a concentration of about 14 mM,
near where the reversal frequency peaks and the CW bias
approaches 0.5.
A Model Mechanism. We assume that the behavior of the
motor can be described by a thermal isomerization model,
specified by the free-energy diagram shown in the Inset of Fig. 4,
where GCCW, GCW, and GT are the free energies of the CCW,
CW, and transition states, respectively [similar to the model
proposed by Khan and Macnab (33)]. The free-energy difference
DG 5 GCCW 2 GCW determines the equilibrium rotational bias,
whereas the activation energies DG‡1 5 GT 2 GCCW and DG‡2 5
GT 2 GCW set the transition rates k1 and k2, respectively. Binding
of CheY** shifts GCCW upwards and GCW downwards, altering
bias and transition rates. We assume that this binding occurs by
mass action to several, independent sites (presumably on different
molecules of FliM; see refs. 34 and 35) and that the occupancy of
any site changes the free energies by a fixed amount. More
specifically, let the binding of n molecules of CheY** raise GCCW
by np, lower GCW by nq, and decrease their energy difference by
nr, where r 5 p 1 q are constants. If the value for DG in the
absence of binding is DGo, then
DG 5 DG o 2 nr
[1]
A value for DGo of 14.4 kT is obtained from a linear extrapolation (to 23°C) of the free-energy vs. temperature plot of
Turner et al. (figure 5C of ref. 31), where we express DGo in
units of energy per motor rather than energy per mole, and kT
is Boltzmann’s constant times absolute temperature, evaluated
(by convention) at 289°K.
If M is the number of binding sites per motor and C is the
concentration of free CheY**, then by mass action the number
of bound sites, n, is
n 5 M@Cy~C 1 K D!#,
[2]
Cell Biology: Scharf et al.
Proc. Natl. Acad. Sci. USA 95 (1998)
205
FIG. 4. Fit to Eq. 3 for DGo 5 14.4 kT, yielding Mr 5 23.1 kT, and
KD 5 9.1 mM. (Inset) The free-energy diagram.
FIG. 3. Behavioral effects of induction of CheY**. The circle
values were obtained by measuring the bias and reversal rate for each
cell, computing k1 and k2 for each cell, and taking the means (and
standard errors) over the cell population (at least 30 cells at each level
of induction). The 3 values were obtained from exponential fits to
interval distributions (where possible), computing the bias and reversal
frequency for each cell, and taking the means over the cell population
(with fewer than 30 cells at low and high bias, where reversal
frequencies were infrequent). The dashed line values were obtained
from fits of the model mechanism to k1 and k2 (Fig. 4). The solid line
values were obtained from fits of the model mechanism to CW bias and
reversal frequency. See the text. (Inset) A segment of an angular speed
record spanning 10 sec; CCW is 1.
where KD is the dissociation constant for the CheY** binding
site. The quantity in square brackets is the fraction of binding
sites occupied (or the probability that a given site is occupied).
The ratio of the probabilities of being in the CW or CCW state,
(CW bias)y(1 2 CW bias) 5 k1yk2, is equal to the Boltzmann
factor exp(2DGykT). Therefore,
ln ~k 1yk 2! 5 2 DG oykT 1 @MrykT# @Cy~C 1 K D!#. [3]
If this equation is fit to the data points of Fig. 3C, given DGo
5 14.4 kT, we obtain Mr 5 23.1 kT, and KD 5 9.1 mM (Fig. 4).
We have assumed that C, the concentration of free CheY**,
is the same as the total amount of CheY**, i.e., that the total
number of binding sites in the cell is small compared with the
total number of molecules of CheY**. The abscissa in Fig. 4
is the fraction of binding sites occupied, Cy(C 1 KD).
From transition-rate theory (see ref. 31) it follows that ln k1 is
equal to a constant plus [MpykT] [Cy(C 1 KD)], and ln k2 is equal
to a different constant minus [MqykT] [Cy(C 1 KD)]. For the
value of KD determined above, these fits yield Mp 5 12.5 kT and
Mq 5 10.6 kT. As expected, Mp 1 Mq 5 Mr. Because p is not very
different from q, binding of CheY** destabilizes the CCW state
by roughly the same degree that it stabilizes the CW state. If there
are, say, 26 CheY** binding sites (36, 37), then the shift in
activation energy per CheY** bound, p or q, is about 0.4 kT, a
value substantially smaller than the change in activation energy
effected by a typical enzyme. Given these fits, the values for k1
and k2 can be plotted again as a function of the concentration of
CheY**, as shown by the dashed lines in Fig. 3C. The corresponding values for CW bias and reversal frequency are shown by
the dashed lines in Fig. 3 A and B. The fit for the reversal
frequency is not very good, but the reason for this lies with
cell-to-cell scatter. The points shown in Fig. 3 are averages over
the cell population of each parameter measured or computed for
each cell separately. One gets different values for CW bias and
reversal frequency when these values are computed from the
average values for k1 and k2, or alternatively, one gets different
values for k1 and k2 when these values are computed from the
average values of CW bias and reversal frequency. The latter fits
are shown by the solid lines in Fig. 3 (for Mr 5 21.1 kT and KD
5 6.4 mM). The fit shown by the solid line in Fig. 3A was obtained
from the bias data alone, because bias is an equilibrium property,
and k1yk2, Eq. 3, is equal to (CW bias)y(1 2 CW bias). However,
the fits shown by the solid lines in Fig. 3 B and C require, in
addition, measurements of rates (the data in Fig. 3B). Our
estimates of Mr and KD are probably no better than about 610%
and 640%, respectively.
DISCUSSION
There are two kinds of mechanisms by which the binding of
molecules of CheY (CheY-P or CheY**) might affect the direction of rotation of a flagellar motor: one deterministic and the
other stochastic. In a deterministic mechanism, the direction of
rotation depends, at any moment in time, on the amount of CheY
bound. Thus, in a two-state model considered (and rejected) by
Kuo and Koshland (5), the motor spins CCW when no CheY is
bound, or CW when one CheY is bound. In this special case,
binding and rotational states are equivalent. More generally, in a
deterministic mechanism, state refers to a binding state, not a
206
Cell Biology: Scharf et al.
rotational state. Thus, for example, in the five-state model
considered by Bray et al. (38), the motor spins CCW when zero
or one CheY is bound, or CW when three or four CheYs are
bound. It is asserted that the sigmoidal nature of the bias plot
reflects cooperative binding, and this is built into the model to
achieve a fit (5, 38). A Hill plot of our data of Fig. 3A has a slope
of 4.2 (not shown).
In a stochastic mechanism, on the other hand, the binding
of CheY determines only the probability of CW or CCW
rotation. For any given fraction of sites bound, switching can
still occur; it is driven by thermal fluctuations. Here, state
refers to direction of rotation, not to CheY binding. The
deterministic model asserts that CheY throws the switch. The
stochastic model asserts, merely, that CheY changes the stabilities of the two rotational states.
One argument in favor of a stochastic mechanism that also
provides justification for a theory involving hopping between
states is that switching events are rare (occur on a time scale that
is long compared with periods of molecular vibration; ref. 33) and
distributions of waiting times (of CCW or CW rotation intervals)
are exponential (39). A multistate deterministic model does not
generally have the latter property, particularly if the rate constants coupling different states have similar orders of magnitude.
This is evident, for example, in figure 4 of Bray et al. (38), which
shows a run-length distribution with a long tail. Another argument in favor of a stochastic model is that transitions can be
induced without any CheY binding at all: rotation in a cell devoid
of CheY can be shifted from exclusively CCW to predominantly
CW simply by lowering the temperature (31).
As evident in Fig. 4, our behavioral data can be fit by a
stochastic model remarkably well, given two simple assumptions: (i) that CheY binds to a set of identical, independent
sites, and (ii) that this binding shifts the energy level of the
CCW state up and the energy level of the CW state down by
amounts directly proportional to the number of molecules
bound. In such a model, motor switching rates do not depend
on CheY-binding off-rates. For a KD in the micromolar range,
one expects binding-site dwell times to be relatively short. If
the binding site on FliM is approximated as a sticky patch of
radius s 5 1 nm and the on-rate for binding is assumed to be
diffusion-limited, then the mean dwell time of CheY on the
binding site will be (4DsKD)21—see ref. 40—where D is the
diffusion coefficient for CheY (about 1027 cm2ys), and KD 5
5.4 3 1015 moleculesycm3 (9.1 mM). This yields a dwell time of
4.6 ms. If this is so, CheY will visit each binding site about 220
times a second or any one of 26 independent sites 5,700 times
a second, i.e., at frequencies much higher than the reversal
frequency. As a consequence, the free-energy difference between CW and CCW states will fluctuate, but the averages
over rotation intervals and thus, the probabilities of hopping
from one state to the other, will remain well defined.
Kuo and Koshland (5), working with CheYwt weakly activated by a CheAyZ fusion protein, obtained CW bias and k1,
k2 curves of approximately the same shape as those shown in
Fig. 3. This equality is expected if CheY-P and CheY** have
similar effects on the motor and if the same fraction of CheYwt
is phosphorylated at different CheYwt concentrations. In
addition, the values of k1 and k2 obtained in the two sets of
experiments for a given bias, say 0.5, are approximately the
same, even though the KD for binding of CheY-P must be
substantially smaller than that for CheY**. This implies that
the motor-switching rates are the same in the two sets of
experiments, even though the dwell times of CheY on the
binding sites are different. As noted above, this is expected for
a stochastic model. It is not expected for a deterministic one.
Finally, our model is similar to one proposed by Macnab (41)
Proc. Natl. Acad. Sci. USA 95 (1998)
but without the assumption that the binding of CheY is highly
cooperative. We have shown that one can obtain a sigmoid bias
curve that displays sizable gain over a narrow concentration
range (10–20 mM for CheY**, Fig. 3A) without that condition.
We thank Fricke Pietruschka for preparing hybridoma cell lines,
Richard Berry for the tethered-cell analysis program, Aravi Samuel for
theoretical insights, and David DeRosier and Karel Svoboda for
comments on the manuscript. B.E.S. was a recipient of an Otto Hahn
Prize Fellowship from the Max Planck Society. This work was supported by National Institutes of Health Grant AI16478 and by the
Rowland Institute for Science.
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